Theorem 0nnq 10335

Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
0nnq	⊢ ¬ ∅ ∈ Q

Proof of Theorem 0nnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5553	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 10323	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3	2	ssrab3 4008	. . 3 ⊢ Q ⊆ (N × N)
4	3	sseli 3911	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
5	1, 4	mto 200	1 ⊢ ¬ ∅ ∈ Q

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2111  ∀wral 3106  ∅c0 4243   class class class wbr 5030   × cxp 5517  ‘cfv 6324  2^nd c2nd 7670  Ncnpi 10255   <_N clti 10258   ~_Q ceq 10262  Qcnq 10263

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5093  df-xp 5525  df-nq 10323

This theorem is referenced by:  adderpq  10367  mulerpq  10368  addassnq  10369  mulassnq  10370  distrnq  10372  recmulnq  10375  recclnq  10377  ltanq  10382  ltmnq  10383  ltexnq  10386  nsmallnq  10388  ltbtwnnq  10389  ltrnq  10390  prlem934  10444  ltaddpr  10445  ltexprlem2  10448  ltexprlem3  10449  ltexprlem4  10450  ltexprlem6  10452  ltexprlem7  10453  prlem936  10458  reclem2pr  10459